Question about automorphisms of ultrapowers

[Noah Schweber]

Hello, all,

I have a seemingly-basic question about ultrapowers which I can't find an
answer to. I've asked on a couple forums (e.g. at mathoverflow:
https://mathoverflow.net/questions/408178/non-rigid-ultrapowers-in-mathsfzfc)
without success; I'm curious if anyone here knows the answer.

Work in ZFC. Suppose A is a countable structure in a countable language.
Must there be an ultrafilter U on the naturals such that the ultrapower of
A by U has a nontrivial automorphism?

An affirmative answer is easy to get if we either assume CH or drop the
requirement that U be an ultrafilter on the naturals specifically.

Thanks,




 - Noah Schweber
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